\begin{answer}
    We have
    $$
    \frac{\partial }{\partial \eta}\int p(y;\eta)dy = 0
    $$

    and 
    $$
    \begin{aligned}
\frac{\partial }{\partial \eta}\int
p(y;\eta)dy &= \int\frac{\partial}{\partial \eta }p(y;\eta)dy\\
&= \int (y - a'(\eta))b(y) \exp(\eta y - a(\eta))dy\\
&=\mathbb E_y[y|\eta] - a'(\eta)
\end{aligned}
$$

So
$$
    a'(\eta) = \mathbb E[y|\eta]
    $$
\end{answer}
